I need help, I am trying to create a formula to calculate the height ($H$) for a bandsaw design.
- The blade length ($L$) shown in red color, desired cutting depth (W), and wheel diameters $A$, $B$, $C$, $D$ will all be known.
- The outer edge of the wheels will always align outward horizontally and inward vertically (see 'Cutting Area').
- $A$, $B$, $C$ diameters will be the same diameter with each other and are all smaller than $D$.
I appreciate the help, I am stuck on this one.
First, take a look at the attached picture annotated.
Firstly, the inclination angle $a$ can be calculated from the difference in diameters of wheels $A$ and $D$.
$$\sin(a) = \frac{D}{W-D-A},$$ where I use the capital letters to represent the wheel labels and also their radii.
Next, compute $b = \pi-(\pi/2-a) = \pi/2+a$. Then, the length of the arc around $D$ is given by $$\text{arc}_{D} = D \times b.$$
Now, as we move from segment $CD$ to segment $AB$ (counter-clockwise), we traverse a total angle of $\pi$. Therefore, $$c = \pi - b = \pi - (\pi/2 + a) = \pi/2-a.$$ And, the arc around this wheel is $$\text{arc}_{A} = A \times (\pi/2-a).$$
Finally, the diagonal segment between $A$ and $D$ is $$\text{diag} = \sqrt{(W-A-D)^2-D^2}.$$
Therefore, the total belt distance is $$L = 2H+(W-B-C) + \pi/2 C + \pi/2 B + \text{arc}_{A} + \text{arc}_{D} + A + D + \text{diag}.$$